\documentclass[]{report}
\usepackage[a4paper, top=3cm, bottom=3cm]{geometry}
\usepackage{amsmath}
\usepackage{color}

%\newcommand{}{{}}

% Turn off all indenting:
\setlength{\parindent}{0cm}

% Constants
\newcommand{\SpeedOfLight}{{c}}
\newcommand{\Xdir}{{\hat{i}}}
\newcommand{\Ydir}{{\hat{j}}}
\newcommand{\Zdir}{{\hat{k}}}

% Modifiers
\newcommand{\Incident}{{_{i}}}
\newcommand{\Reflected}{{_{r}}}
\newcommand{\Transmitted}{{_{t}}}
\newcommand{\Vacuum}{{_{0}}}
\newcommand{\Maximum}{{_{max}}}
\newcommand{\Minimum}{{_{min}}}


% Wave Parameters
\newcommand{\Frequency}{{\nu}}
\newcommand{\Wavelength}{{\lambda}}
\newcommand{\Period}{{\tau}}
\newcommand{\Velocity}{{v}}
\newcommand{\Perturbance}{{u}}
\newcommand{\Amplitude}{{A}}
\newcommand{\AngularFrequency}{{\omega}}
\newcommand{\WaveNumber}{{k}}
\newcommand{\WaveNumberVec}{{\vec{k}}}
\newcommand{\InitialPhase}{{\phi_{0}}}
\newcommand{\Intensity}{{I}}
\newcommand{\EField}{{\vec{E}}}
\newcommand{\EFieldAmplitude}{{\vec{E_{0}}}}
\newcommand{\EFieldMag}{{E}}
\newcommand{\HField}{{\vec{H}}}
\newcommand{\HFieldAmplitude}{{\vec{H_{0}}}}
\newcommand{\BField}{{\vec{B}}}
\newcommand{\BFieldAmplitude}{{\vec{B_{0}}}}


% Inter-Wave Parameters
\newcommand{\TotalPhaseDifference}{{\delta_{tot}}}
\newcommand{\PhaseDifference}{{\delta}}
\newcommand{\Contrast}{{C}}

% Medium Properties
\newcommand{\Permittivity}{{\epsilon}}
\newcommand{\Permeability}{{\mu}}
\newcommand{\Impedance}{{\eta}}
\newcommand{\Absorbtion}{{\alpha}}
\newcommand{\Abbe}{{\nu_{D}}}

% Geometric Properties
\newcommand{\RefractiveIndex}{{n}}
\newcommand{\OpticalPathLength}{{L}}

% General Variables
\newcommand{\Time}{{t}}
\newcommand{\Direction}{{r}}
\newcommand{\DirectionVec}{{\vec{r}}}
\newcommand{\Angle}{{\theta}}
\newcommand{\Radius}{{r}}
\newcommand{\Integer}{{n}}
\newcommand{\Phase}{{\Phi}}
\newcommand{\PointSeparation}{{a}}
\newcommand{\Distance}{{D}}
\newcommand{\Thickness}{{d}}
\newcommand{\Finesse}{{F}}
\newcommand{\ReflectanceCoeff}{{r}}
\newcommand{\Width}{{w}}
\newcommand{\NumberOfSlits}{{N}}


\title{Optics and Photonics Study Guide}
\author{Alexandra Booth \and Glenn Sweeney}
\date{16 January 2014}

\begin{document}

\maketitle
\newpage

\tableofcontents

\chapter{Physical Optics}

\section{Variable Definitions}

\subsection{Constants}

$\SpeedOfLight$: Speed of Light - $3\cdot10^{8}\frac{m}{s}$ in vacuum.

$\Xdir, \Ydir, \Zdir$: Euclidian basis vectors - each vector points in x, y, z, respectively, with unit length.

\subsection{Wave Properties}

$\Perturbance$: Perturbance - a generic term that oscillates as a harmonic wave.

$\Wavelength$: Wavelength - The distance covered by one period of the wave through space.

$\Frequency$: Temporal Frequency - The number of oscillations per second of a wave.

$\Period$: Period - The time it take a wave to complete an oscillation.

$\Velocity$: Velocity - The speed of propogation of a wave.

$\AngularFrequency$: Angular Frequency - The temporal frequency of the wave expressed in radians.

$\WaveNumber$: Wave Number - The spatial frequency of the wave expressed in radians.

$\WaveNumberVec$: Wave Number (vector) - A vector of magnitude $\WaveNumber$ that points in the direction of propogation of a wave.

$\InitialPhase$: Initial Phase - The phase of a wave at the origin of the coordinate system.

$\Phase$: Phase - The fraction of the wave cycle relative to the origin.

$\Amplitude$: Amplitude - The magnitude from the peak to the trough of a wave.

$\EFieldMag$: Electric Field Magnitude- The scalar value of the magnitude of the electric field as a function of the wave parameters.

$\EField$: Electric Field Vector - A vector pointing in the direction of the electric field oscillation. The magnitude represents the strength of the electric field.

$\EFieldAmplitude$: Electric Field Amplitude Vector - A vector denoting the direction and magnitude of the amplitude of the electric field.

$\HField$: Magnetic Field Vector -A vector pointing in the direction of the magnetic field oscillation. The magnitude represents the strength of the magnetic field.

$\HFieldAmplitude$: Magnetic Field Amplitude Vector - A vector denoting the direction and magnitude of the amplitude of the magnetic field.

$\BField$: Magnetic Flux Density Vector -A vector pointing in the direction of the magnetic flux density (in line with the magnetic field vector). The magnitude represents the strength of the magnetic flux density.

$\BFieldAmplitude$: Magnetic Flux Density Amplitude Vector - A vector denoting the direction and magnitude of the amplitude of the magnetic flux density.

$\Intensity$: Intensity - The measured quantity of electromagnetic radiation.

\subsection{Inter-Wave Properties}

$\TotalPhaseDifference$: Total Phase Difference - The total phase difference at a given point between two interfering waves.

$\PhaseDifference$: Phase Difference - Any measured or calculated phase difference between interfering waves.

$\Contrast$: Constrast - A term that denotes the visibility of an interference pattern.

$\Finesse$: Finesse - A term used to measure the behavior of certain interferometric systems.

\subsection{Medium Properties}

$\RefractiveIndex$: Refractive Index - A measure of the deviation of light as it enters or leaves a material.

$\Permittivity$: Dielectric Permittivity - The resistance encountered by an electric field in a medium.

$\Permeability$: Magnetic Permeability - The ability of a medium to support the formation of a magnetic field.

$\Impedance$: Impedance - A calculated property of a material.

$\Absorbtion$: Coefficient of Absorbtion - A measured coefficient that determines the decrease of electromagnetic field amplitude in a material.

$\Abbe$: Abbe - A measure of a material's chromatic dispersion.

$\ReflectanceCoeff$: Reflectance Coefficient - A measure of the percentage of electric field amplitude reflected by an interface.


\subsection{Geometric Properties}

$\Time$: Time - a variable that denotes time elapsed from the origin.

$\Direction$: Direction - A variable that denotes the magnitude of distance from the origin.

$\DirectionVec$: Direction Vector - A variable that denotes the magnitude and direction of distance rom the origin.

$\Angle$: Angle - A generic constant representing the angle subtending two lines.

$\Radius$: Radius - A measure of the size of a circle or sphere.

$\PointSeparation$: Point Separation - A measure of the distance between two points on the vertical axis.

$\Distance$: Distance - a measure between two points or planes on the horizontal axis.

$\Thickness$: Thickness - A measure of the space covered by a medium.

$\Width$: Width - A generic measure of a dimension of an object.

$\OpticalPathLength$: Optical Path Length - Distance traveled, as seen by a propogating wave.

$\NumberOfSlits$: Number of Slits - The number of slits considered on a diffraction grating.

\subsection{General Variables}

$\Integer$: A generic filler number to satisfy periodic relations. It must be non-fractional.


\section{Electromagnetic Radiation}

\subsection{General Wave Properties}

Perturbance $(\Perturbance)$ is a function of space and time:
\begin{eqnarray}
\Perturbance &=& f(\DirectionVec, \Time) \\
\Perturbance &=& f(\DirectionVec) \pm f(\Time)
\end{eqnarray}
Where:
\begin{eqnarray}
\DirectionVec = x\Xdir + y\Ydir + z\Zdir
\end{eqnarray}
is a position in 3 dimensional space. \\

If the angular frequency $(\AngularFrequency)$ is negative, the wave travels in positive $\WaveNumberVec$ direction in time.

If it is positive, the wave travels in negative $\WaveNumberVec$ direction in time. \\

General form of the wave equation in three dimensions:
\begin{equation}
\frac{\delta^{2}u}{\delta x^{2}} + \frac{\delta^{2}u}{\delta y^{2}} + \frac{\delta^{2}u}{\delta z^{2}} = \frac{1}{\Velocity ^{2}} \cdot \frac{\delta^{2}u}{\delta t^{2}}
\end{equation}

A wave propogates in a direction indicated by the vector $(\WaveNumberVec)$.
\begin{equation}
\WaveNumberVec = \WaveNumber _{x} + \WaveNumber _{y} + \WaveNumber _{z}
\end{equation}


\subsection{Wave Parameter Transforms}

\begin{equation}
\WaveNumber = |\WaveNumberVec|
\end{equation}

\begin{equation}
\Period = \frac{1}{\Frequency}
\end{equation}

\begin{equation}
\Frequency = \frac{\Velocity}{\Wavelength} = \frac{\SpeedOfLight}{\Wavelength \Vacuum}
\end{equation}

\begin{center}
Remember, frequency $(\Frequency)$ is always constant regardless of medium!
\end{center}

\begin{equation}
\AngularFrequency = 2\pi \Frequency
\end{equation}

\begin{equation}
\label{eqn:krelation}
\WaveNumber = \frac{2\pi}{\Wavelength}
\end{equation}

\begin{equation}
\text{Non-Essential: }\Velocity = \frac{\AngularFrequency}{\WaveNumber}
\end{equation}

\subsubsection{Medium-Dependent Transforms}

\begin{equation}
\RefractiveIndex = \frac{\SpeedOfLight}{\Velocity}
\end{equation}

\begin{equation}
\text{Non-Essential: }\Wavelength = \frac{\Wavelength \Vacuum}{\RefractiveIndex}
\end{equation}

\subsection{Planar Harmonic Wave}

Primary (general) form:
\begin{equation}
\Perturbance(\DirectionVec, \Time) = \Amplitude sin\left( \WaveNumberVec \cdot \DirectionVec - \AngularFrequency \Time + \InitialPhase \right) \\
\end{equation}

Note: if $\WaveNumberVec$ is along the $z$ axis, then $\WaveNumberVec \cdot \DirectionVec = kz$. \\

Other forms (with $\WaveNumberVec$ along the $z$ axis):
\begin{eqnarray}
\Perturbance(z, \Time) &=& \Amplitude sin\left( 2\pi \left( \frac{z}{\Wavelength} - \frac{\Time}{\Period}\right) + \InitialPhase \right) \\
\Perturbance(z, \Time) &=& \Amplitude sin\left(k \left( z - \Velocity \Time \right) + \InitialPhase \right)
\end{eqnarray}

Note: ``Phase'' is the entire term inside the sine function! \\

In exponential form:
\begin{equation}
\Perturbance(\DirectionVec, \Time) = \Re \left( Ae^{i(\WaveNumberVec \cdot \DirectionVec - \AngularFrequency \Time + \InitialPhase)} \right)
\end{equation}

\subsection{Spherical Harmonic Wave}

Primary (general) form:
\begin{equation}
\Perturbance(\Radius, \Time) = \frac{\Amplitude}{\Radius} sin\left( \WaveNumber \Radius - \AngularFrequency \Time + \InitialPhase \right) 
\end{equation}

Note that in this case, $\Direction$ and $\WaveNumber$ are scalars because the wave is isotropic. \\


In exponential form:

\begin{equation}
\Perturbance(\Radius, \Time) = \Re \left( \frac{\Amplitude}{\Radius}e^{i(\WaveNumber \Radius - \AngularFrequency \Time + \InitialPhase)} \right)
\end{equation}

\subsection{Principle of Superposition}

\begin{equation}
\Perturbance(\DirectionVec, \Time) = \Perturbance_{1}(\DirectionVec, \Time) + \Perturbance_{2}(\DirectionVec, \Time)
\end{equation}

\begin{equation}
\Perturbance(\DirectionVec, \Time) = \sum_{j} \Perturbance_{j}(\DirectionVec, \Time)
\end{equation}

\subsection{More Properties of Media}

% Index of Refraction (again)
\begin{equation}
\RefractiveIndex = \sqrt{\frac{\Permittivity}{\Permittivity \Vacuum} \frac{\Permeability}{\Permeability \Vacuum}}
\end{equation}

% Relative Permittivity
\begin{equation}
\Permittivity_{r} = \frac{\Permittivity}{\Permittivity \Vacuum}
\end{equation}

% Relative Permeability
\begin{equation}
\Permeability_{r} = \frac{\Permeability}{\Permeability \Vacuum}
\end{equation}

For nonmagnetic material, $\Permeability_{r} = 1$, and therefore

\begin{equation}
\RefractiveIndex = \sqrt{\frac{\Permittivity}{\Permittivity \Vacuum}}
\end{equation}

\subsection{Maxwell's Wave Equation}

Maxwell's Equations apply for Dielectric, Nondispersive, Homogeneous, Linear, Isotropic media. \\

{\color{red}I can haz cross product...?}

%Condensed Maxwell's
\begin{equation}
\WaveNumberVec \times \EFieldAmplitude = \AngularFrequency \Permeability \HFieldAmplitude
\end{equation}

\begin{equation}
|\BFieldAmplitude| = \Permeability |\HFieldAmplitude|
\end{equation}

\begin{equation}
|\BFieldAmplitude| = \frac{|\EFieldAmplitude|}{\SpeedOfLight}
\end{equation}

{\color{red} Does this relationship only hold true for a vacuum? Should the speed of light more generally be $\Velocity$, because these forms are dependent on $\WaveNumberVec$?}

$\EField$ and $\HField$ ALWAYS have the same {\it frequency}.

$\BField$ and $\HField$ ALWAYS point in the same {\it direction}. \\

A wave equation of $\Perturbance$ can describe $\EField$ with amplitude $\EFieldAmplitude$ or $\HField$ with amplitude $\HFieldAmplitude$. \\

K = pointer

E = middle finger

H = Thumb

\subsection{Intensity}

\begin{equation}
\Impedance = \sqrt{\frac{\Permeability}{\Permittivity}}
\end{equation}

General Expression for Intensity (Exponential Form)
\begin{equation}
\Intensity = \frac{1}{2 \Impedance}|\EField|^{2} = \frac{1}{2 \Impedance}\EField \cdot \EField^{*}
\end{equation}

General Expression for Intensity (Sinusoidal Form)
\begin{equation}
\Intensity = \frac{1}{\Impedance} \langle \EField^{2}\rangle_{T}
\end{equation}

Specific Expression for Intensity of Plane Wave (Any Form)
\begin{equation}
\Intensity = \frac{1}{2 \Impedance} \EFieldAmplitude^{2}
\end{equation}

Specific Expression for Intensity of Spherical Wave (Any Form)
\begin{equation}
\Intensity = \frac{1}{2 \Impedance}\frac{\EFieldAmplitude^{2}}{\Radius^{2}}
\end{equation}

\subsection{Coherence}
A lack of coherence is the final state of entropy for radiation.

The less coherent the light is, the lower the contrast of interference will be. 

Light is naturally incoherent, unless it is a laser source, or an interferometer is used to divide a wave to artificially create coherent light.

\section{Polarization}

\subsection{Mathematical Description}

All equations in this section assume $\WaveNumberVec$ along the $z$ axis. \\

Definitions of $x$ and $y$ components of an electric field vector $\EField$:

\begin{equation}
\EField = \EField_{x} + \EField_{y}
\end{equation}

\begin{eqnarray}
\EField_{x}(z, t) &=& \EFieldAmplitude_{x} cos \left(\WaveNumber z - \AngularFrequency \Time \right) \\ \nonumber \\
\EField_{y}(z, t) &=& \EFieldAmplitude_{y} cos \left(\WaveNumber z - \AngularFrequency \Time + \PhaseDifference \right) 
\end{eqnarray}


\begin{equation}
\PhaseDifference = \Phase_{y} - \Phase_{x}
\end{equation}

General equation for elliptical polarization:
\begin{equation}
{\frac{\EField_{y}}{\EFieldAmplitude_{y}}}^{2} + {\frac{\EField_{x}}{\EField_{x}}}^{2} - 2 \frac{\EField_{y}}{\EFieldAmplitude_{y}} \frac{\EField_{x}}{\EFieldAmplitude_{x}}cos(\PhaseDifference) = sin^{2}(\PhaseDifference)
\end{equation}

Left polarized (counterclockwise in time) is for $0 < \PhaseDifference < \pi$.

Right polarized (clockwise in time) is for $-\pi < \PhaseDifference < 0$. \\

Linear at $\PhaseDifference = 0$, $\PhaseDifference = \pi$.

Circular at $\PhaseDifference = \frac{\pi}{2}$, $\PhaseDifference = -\frac{\pi}{2}$.

\subsection{Birefringence}

Anisotropic crystals have different indices of refraction depending on direction of polarization.

\subsection{Polarizers}

Linear polarizers pass the component of $\EFieldAmplitude$ projected onto the axis of the polarizer, and $\EFieldAmplitude$ emerges on the same axis as the polarizer. \\

Malus' Law:
\begin{equation}
\Intensity \Transmitted = \Intensity \Incident cos^{2}(\Angle)
\end{equation}
where $\Angle$ is the angle between $\EFieldAmplitude$ and the axis of the polarizer.

\section{Classical Interactions of Light and Matter}

\subsection{Absorbtion}

Index of refraction can be expressed as a complex number in order to include information about the absorbtion of a material.
\begin{equation}
\RefractiveIndex = \RefractiveIndex_{0} + i\frac{\Absorbtion}{2 \AngularFrequency}c
\end{equation}

Planar wave equation with absorbtion:
\begin{equation}
\EField = \EFieldAmplitude e^{-\frac{\Absorbtion}{2}z}e^{i\AngularFrequency \left( \frac{\RefractiveIndex_{0}}{c}z - t \right) }
\end{equation}

Glenn and Ali prefer to think of this as:
\begin{equation}
\EField = \EFieldAmplitude e^{-\frac{\Absorbtion}{2}z}e^{i(\WaveNumber_{0}z - \AngularFrequency t) }
\end{equation}
where $\WaveNumber_{0}$ is the wave number corresponding to $\RefractiveIndex_{0}$. \\

Bouger-Beer's Law:
\begin{equation}
\Intensity = \Intensity_{0}e^{-\Absorbtion z}
\end{equation}

All materials have spectral band(s) on which they absorb radiation, and spectral band(s) on which they transmit radiation, due to the resonance frequencies of particles in a medium.

\subsection{Chromatic Dispersion}

$\Permittivity$ is a function of $\Frequency$. Because of this, $\RefractiveIndex$, $\Wavelength$, and $\Velocity$ are also functions of $\Frequency$.

\begin{eqnarray}
\Abbe &=& \frac{\RefractiveIndex_{D} - 1}{\RefractiveIndex_{F} - \RefractiveIndex_{C}} \\
\text{Where:} \\
\RefractiveIndex_{F} &=& \RefractiveIndex(486.1 nm) \\
\RefractiveIndex_{D} &=& \RefractiveIndex(589.2 nm) \\
\RefractiveIndex_{C} &=& \RefractiveIndex(656.3 nm)
\end{eqnarray}

Crown Material: $\Abbe > 50$

Flint Material: $\Abbe < 50$ \\

{\color{red} Information about absorbtion and refractive index plots need to go here still.}

\subsection{Scattering}

Rayleigh Scattering: Small particles, scattering is dependant on wavelength only.
Mie Scattering: Particles size on the order of wavelength.

Raman Scattering: Wavelength shift is introduced during the scattering of the initial wave.

\section{Interference and Diffraction}

Add electric fields if light is coherent, NOT intensities! \\

Constructive Interference: $\TotalPhaseDifference = 2\Integer \pi$

Destructive Interference: $\TotalPhaseDifference = (2\Integer -1)\pi$

\begin{equation}
\label{eqn:generalphasedifference}
\PhaseDifference = \frac{2\pi}{\Wavelength}\OpticalPathLength = \WaveNumber \OpticalPathLength
\end{equation}

\subsection{Planar Wave Interference}

\begin{equation}
\label{eqn:interferenceintensity}
\Intensity = \Intensity_{1} + \Intensity_{2} + 2\sqrt{\Intensity_{1}\Intensity_{2}}(cos(\TotalPhaseDifference))
\end{equation}

\begin{equation}
\TotalPhaseDifference = (\WaveNumberVec_{1} - \WaveNumberVec_{2})\DirectionVec + \InitialPhase_{1}-\InitialPhase_{2}
\end{equation}

\begin{eqnarray}
\Intensity \Maximum &=& \Intensity_{1} + \Intensity_{2} + 2\sqrt{\Intensity_{1}\Intensity_{2}} \\
\Intensity \Minimum &=& \Intensity_{1} + \Intensity_{2} - 2\sqrt{\Intensity_{1}\Intensity_{2}}
\end{eqnarray}

\begin{equation}
C = \frac{\Intensity \Maximum - \Intensity \Minimum}{\Intensity \Maximum + \Intensity \Minimum} = \frac{2\sqrt{\Intensity_{1}\Intensity_{2}}}{\Intensity_{1} + \Intensity_{2}}
\end{equation}

{\color{red} Review of Unit Vectors and Vector Magnitudes} \\

The propogation of fringes for two intersecting plane waves is described by $(\WaveNumberVec_{1} - \WaveNumberVec_{2})$. The period of this propogation can be found using Equation \ref{eqn:krelation}.

\subsection{Spherical Wave Interference}

Same as planar, but:

\begin{equation}
\TotalPhaseDifference = (\DirectionVec_{1} - \DirectionVec_{2})\WaveNumber + \InitialPhase_{1} - \InitialPhase_{2}
\end{equation}
However, this form is ugly! (hyperboloid functions are gross, man.) So, for a long distance away, close to the axis, with points symmetric to the $z$ axis, we approximate spherical waves as plane waves and get:

\begin{equation}
\label{eqn:planewaveapprox}
\TotalPhaseDifference = \frac{ \WaveNumber \PointSeparation x }{\Distance} + \InitialPhase_{1} - \InitialPhase_{2}
\end{equation}



\subsection{Interferometry}

\subsubsection{Young's Double Slit}

For the Young's Double Slit, the general form is the same as Equation \ref{eqn:interferenceintensity}, as two quasiplane waves are interfering. However, due to the construction of the problem, $\Intensity_{1} = \Intensity_{2}$. As a result, it may be rewritten as:

\begin{equation}
\Intensity = 2\Intensity_{0}\left(1+cos(\TotalPhaseDifference)\right)
\end{equation}

If $\InitialPhase_{1} = \InitialPhase_{2}$ (in $\TotalPhaseDifference$) this may be simplified using trigonometric identities to:

\begin{equation}
\Intensity = 4 \Intensity_{0}cos^{2}\left(\frac{\WaveNumber \PointSeparation x}{2\Distance} \right)
\end{equation}

In the case where $\InitialPhase_{1} \ne \InitialPhase_{2}$, the initial phases may be determined geometrically by implementing Equation \ref{eqn:generalphasedifference}.

\begin{equation}
\Wavelength_{fringe} = \frac{\Wavelength \Distance}{\PointSeparation}
\end{equation}
Where $\Wavelength_{fringe}$ is the period of the fringes on a perpindicular target.

Polychromatic Source:

\begin{equation}
\Intensity = \Intensity_{0}(1+sinc\left( \frac{ax(\Frequency_{2}-\Frequency_{1})}{\SpeedOfLight \Distance}\right) cos \left( \frac{2\pi \PointSeparation x \Frequency_{0}}{\SpeedOfLight \Distance}\right)
\end{equation}

where:

\begin{equation}
\Frequency_{0} = \frac{\Frequency_{2} + \Frequency_{1}}{2}
\end{equation}


\subsubsection{Thin Films}

\color{magenta}

\begin{equation}
\PhaseDifference = 2\WaveNumber \RefractiveIndex \Thickness( cos(\Angle \Transmitted))
\end{equation}

\begin{equation}
\Finesse = \left(\frac{2\ReflectanceCoeff}{1-\ReflectanceCoeff^{2}}\right)^{2}
\end{equation}

\begin{equation}
\Intensity \Transmitted = \Intensity_{0}\frac{1}{1+\Finesse sin^{2}\left(\frac{\PhaseDifference}{2}\right)}
\end{equation}

\begin{equation}
\Intensity \Reflected = \Intensity_{0}\frac{\Finesse sin^{2}\left(\frac{\PhaseDifference}{2}\right)}{1+\Finesse sin^{2}\left(\frac{\PhaseDifference}{2}\right)}
\end{equation}

\color{black}

For our case, assume normal incidence. Thus:

\begin{equation}
\PhaseDifference = 2\WaveNumber \RefractiveIndex \Thickness
\end{equation}

{\it Transmissive} maxima exist at $\PhaseDifference = 2n\pi$. In this case, the {\it Reflection} is a minimum.

Opposite for $\PhaseDifference = (2n-1)\pi$.

\subsubsection{Slit Aperture}

\begin{equation}
\EFieldMag \propto sinc\left(\frac{\Width sin(\Angle)}{\Wavelength}\right)
\end{equation}

\begin{equation}
\Intensity \propto sinc^{2}\left(\frac{\Width sin(\Angle)}{\Wavelength}\right)
\end{equation}

where
\begin{equation}
sinc(x) = \frac{sin(\pi x)}{\pi x}
\end{equation}

Remember that when the quantity in the sinc term is set to an integer $n$ (the value in the sine term of the sinc set to $n\pi$), you can determine the dark fringes. Becuase the sinc function converges to 1 for $n=0$, this case is an exception. \\

The result for a rectangular aperture is the same as the result of a slit in the $x$ direction multiplied by the result of a slit in the $y$ direction. \\

\subsubsection{Circular Aperture}
    
\begin{equation}
    \Intensity = \left(\frac{2 J_{1}(\WaveNumber \PointSeparation sin(\Angle))}{\WaveNumber \PointSeparation sin(\Angle)}\right)^{2}
\end{equation}

where $\PointSeparation$ is the diameter of the aperture, and  $J_{1}$ is the Bessel function of the first kind. \\

\subsubsection{Double Slit Revisited}

\begin{equation}
\Intensity = 4\Intensity_{0}sinc^{2}\left(\frac{\Width sin(\Angle)}{\Wavelength}\right) cos^{2}\left(\frac{\WaveNumber \PointSeparation x}{2\Distance}\right)
\end{equation}

In this case, $sin(\Angle) \approx \frac{x}{\Distance}$, because of the small angle approximation and the planar wave approximation. This expression is also proportionally equivalent to the multiplication of single slit diffraction and double slit diffraction.

\subsubsection{Diffraction Gratings}

\begin{equation}
\Intensity = \frac{\Intensity_{0}}{\NumberOfSlits^{2}}sinc^{2}\left(\frac{\Width sin(\Angle)}{\Wavelength}\right)\left( \frac{sin(\NumberOfSlits \PhaseDifference)}{sin(\PhaseDifference)}\right)^{2}
\end{equation}

$\PhaseDifference$, in this case, is defined as:

\begin{equation}
\PhaseDifference = \WaveNumber \frac{\PointSeparation}{2}sin(\Angle)
\end{equation}

Maxima:
\begin{equation}
\PointSeparation (sin(\Angle \Transmitted)- sin(\Angle \Incident)) = n\Wavelength
\end{equation}

The central peak is 2 periods wide.

\chapter{Geometrical Optics}

\newcommand{\Vergence}{{\Phi}}
\newcommand{\ObjectFocalLength}{{f}}
\newcommand{\ImageFocalLength}{{f'}}
\newcommand{\ConvergenceRatio}{{G}}
\newcommand{\ObjectRayAngle}{{u}}
\newcommand{\ImageRayAngle}{{u'}}
\newcommand{\TransverseZoom}{{\gamma_{t}}}
\newcommand{\AxialZoom}{{\gamma_{a}}}
\newcommand{\ObjectDistance}{{p}}
\newcommand{\ImageDistance}{{p'}}
\newcommand{\ObjectExtraFocalDist}{{\Pi}}
\newcommand{\ImageExtraFocalDist}{{\Pi '}}

\section{Variable Definitions}

\subsection{General Definitions}


$n$: Index of refraction left of the interface.

$n'$: Index of refraction right of the interface.

$n_{a}$: Index of refraction in between interfaces.

$C$: Center of curvature of a dioptre.

$S$: Summit of a dioptre.

$O$: The center of an optical system.

$F$: Object focal point.

$F'$: Image focal point.

$\ObjectExtraFocalDist$: Object Extra-Focal Distance.

$\ImageExtraFocalDist$: Image Extra-Focal Distance.

$\ObjectDistance$: Object Distance.

$\ImageDistance$: Image Distance.

\subsection{Spherical Dioptres}
\begin{eqnarray}
\ObjectFocalLength &=& \overline{SF}\nonumber \\
\ImageFocalLength &=& \overline{SF'}\nonumber \\
\ObjectDistance &=& \overline{SA}\nonumber \\
\ImageDistance &=& \overline{SA'}\nonumber \\
\ObjectExtraFocalDist &=& dunno\nonumber \\
\ImageExtraFocalDist &=& dunno\nonumber
\end{eqnarray}
{\color{red} Can I define $\Pi$ and $\Pi'$ for the spherical dioptre?}


\subsection{Thin Lenses}
\begin{eqnarray}
\ObjectFocalLength &=& \overline{OF}\nonumber \\
\ImageFocalLength &=& \overline{OF'}\nonumber \\
\ObjectDistance &=& \overline{OA}\nonumber \\
\ImageDistance &=& \overline{OA'}\nonumber \\
\ObjectExtraFocalDist &=& \overline{FA}\nonumber \\
\ImageExtraFocalDist &=& \overline{F'A'}\nonumber
\end{eqnarray}


\subsection{Optical Systems}
\begin{eqnarray}
\ObjectFocalLength &=& \overline{HF}\nonumber \\
\ImageFocalLength &=& \overline{H'F'}\nonumber \\
\ObjectDistance &=& \overline{HA}\nonumber \\
\ImageDistance &=& \overline{H'A'}\nonumber \\
\ObjectExtraFocalDist &=& \overline{FA}\nonumber \\
\ImageExtraFocalDist &=& \overline{F'A'}\nonumber
\end{eqnarray}



\section{Equations}

\subsection{General Equations}

\begin{equation}
\OpticalPathLength = \int_A^B \! \RefractiveIndex(\DirectionVec) \, \mathrm{\Thickness}s 
\end{equation}

\begin{center}
For homogeneous media:
\end{center}
\begin{equation}
\OpticalPathLength = \RefractiveIndex \Thickness
\end{equation}
Fermat's Principle Mirages Fiber Optic - Bend towards high $\RefractiveIndex$

% Law of Reflection
\begin{equation}
\Angle \Incident = \Angle \Reflected
\end{equation}

% Law of Refraction (Snell's Law)
\begin{equation}
\RefractiveIndex \Incident sin(\Angle \Incident) = \RefractiveIndex \Transmitted sin(\Angle \Transmitted)
\end{equation}

% Total Internal Reflection
When $\Angle \Transmitted = 90^{\circ}$:
\begin{equation}
    \Angle \Incident = \Angle_{c} = sin^{-1}\left( \frac{\RefractiveIndex \Transmitted}{\RefractiveIndex \Incident} \right)
\end{equation}

Abbe Relation:
\begin{equation}
\RefractiveIndex \ObjectRayAngle = \RefractiveIndex '\ImageRayAngle \text{ when } \TransverseZoom = 1
\end{equation}


\begin{equation}
\Vergence = \frac{\RefractiveIndex '}{\ImageFocalLength} = -\frac{\RefractiveIndex}{\ObjectFocalLength}
\end{equation}

In words, vergence is the reciprocal focal length, adjusted for optical path length. Positive vergence describes convergent rays.

Convergence Ratio can also be called Angular Zoom:
\begin{equation}
\ConvergenceRatio = \frac{\ImageRayAngle}{\ObjectRayAngle}
\end{equation}

\begin{equation}
\TransverseZoom = \frac{\overline{A'B'}}{\overline{AB}} = \frac{\RefractiveIndex \ImageDistance}{\RefractiveIndex '\ObjectDistance} = -\frac{\ObjectFocalLength}{\ObjectExtraFocalDist} = -\frac{\ImageExtraFocalDist}{\ImageFocalLength}
\end{equation}

\begin{equation}
\AxialZoom = -\frac{\ImageExtraFocalDist}{\ObjectExtraFocalDist} = \frac{\RefractiveIndex}{\RefractiveIndex '}\TransverseZoom ^{2}
\end{equation}

Lagrange-Helmholtz Relation:
\begin{equation}
\ConvergenceRatio \TransverseZoom = \frac{\RefractiveIndex}{\RefractiveIndex '}
\end{equation}

Newton Relation:
\begin{equation}
\ObjectExtraFocalDist \ImageExtraFocalDist = \ObjectFocalLength \ImageFocalLength
\end{equation}


\subsection{Spherical Dioptres (Interfaces)}
\begin{equation}
\frac{n'}{\ImageDistance} - \frac{n}{\ObjectDistance} = \frac{n'-n}{\overline{SC}}
\end{equation}

For a spherical dioptre, $\overline{SC}$ is the radius of curvature of the diopre.

Focal points may be found using this equation by setting the appropriate object or image distance to $\infty$.

\begin{equation}
\ObjectFocalLength + \ImageFocalLength = \overline{SC}
\end{equation}

\begin{equation}
\frac{\ObjectFocalLength}{\ImageFocalLength} = -\frac{\RefractiveIndex}{\RefractiveIndex '}
\end{equation}

A mirror is a spherical dioptre, where $\RefractiveIndex '=-\RefractiveIndex$.


\subsection{Thin Lenses}

The optical center $O$ is the center point of $S_{1}$ and $S_{2}$

\begin{equation}
\frac{1}{\ImageDistance} - \frac{1}{\ObjectDistance} = \frac{1}{\ImageFocalLength}
\end{equation}

\begin{equation}
\ObjectFocalLength = -\ImageFocalLength \text{ if } \RefractiveIndex = \RefractiveIndex '
\end{equation}

\begin{equation}
(\RefractiveIndex_{a}-1)\left[\frac{1}{\overline{OC_{1}}}-\frac{1}{\overline{OC_{2}}}\right] = \frac{1}{\ImageFocalLength} = -\frac{1}{\ObjectFocalLength} = \Phi
\end{equation}


\subsection{Associated Thin Lenses}


\begin{equation}
\frac{1}{\ImageFocalLength_{1}} + \frac{1}{\ImageFocalLength_{2}} = \frac{1}{\ImageFocalLength_{s}}
\end{equation}

\begin{equation}
\ObjectFocalLength_{s} = -\ImageFocalLength_{s} \text{ if } \RefractiveIndex = \RefractiveIndex '
\end{equation}

\subsection{Optical Systems}


\begin{equation}
\ObjectFocalLength_{s} = -\ImageFocalLength_{s} \text{ if } \RefractiveIndex = \RefractiveIndex '
\end{equation}

\subsection{Associated Optical Systems}

\begin{equation}
\overline{F_{2}'F_{s}'} = -\frac{\ObjectFocalLength_{2}\ImageFocalLength_{2}}{\overline{F_{1}'F_{2}}}
\end{equation}

\begin{equation}
\overline{F_{1}F_{s}} = -\frac{\ObjectFocalLength_{1}\ImageFocalLength_{1}}{\overline{F_{1}'F_{2}}}
\end{equation}

\begin{equation}
\Vergence = \Vergence_{1} + \Vergence_{2} -\frac{e}{\RefractiveIndex_{a}}\Vergence_{1}\Vergence_{2}
\end{equation}

where $\RefractiveIndex_{a}$ is the refractive index between the two optical systems.

\begin{equation}
\overline{H_{2}'H_{s}'} = -\RefractiveIndex '\frac{e}{\RefractiveIndex_{a}}\left(\frac{\Vergence_{1}}{\Vergence_{s}}\right)
\end{equation}


\begin{equation}
\overline{H_{2}H_{s}} = \RefractiveIndex \frac{e}{\RefractiveIndex_{a}}\left(\frac{\Vergence_{2}}{\Vergence_{s}}\right)
\end{equation}

\begin{equation}
e=\overline{H_{1}'H_{2}}
\end{equation}

\begin{equation}
\ObjectFocalLength_{s} = -\ImageFocalLength_{s} \text{ if } \RefractiveIndex = \RefractiveIndex '
\end{equation}

Principal Planes are planes where $\TransverseZoom = 1$.\\

Nodal Points are points through which rays continue undeviated through the system. \\

If $\RefractiveIndex = \RefractiveIndex '$, then the Nodal Points are on the Principal Points.

\section{General Notes}

Stigmatic System - The object and image are perfect, conjuate, reversible points.

Gauss Conditions - Incoming beam angles are small and diffraction is negligible.\\


Real object - rays are diverging.

Virtual object - Rays are converging.

Real image - Rays are converging.

Virtual image - Rays are diverging. \\

Some indices of refraction:

$n_{air} = 1.00$

$n_{water} = 1.33$

$n_{glass} = 1.53$ \\

{\color{red} Copy other equations to here.} \\


\section{Raytracing}

Any ray entering system parallel to axis goes through F'.
Any ray leaving the system parallel to the axis came from F.

The set of rays emerging from a point on the focal plane emerge parallel.

Rays incident on a principal plane emerge with no height change on the other plane.

Rays passing through a nodal point emerge undeviated from the other nodal point

\appendix

\section{Trigonometric Identities}


{\color{red} Learn random trig identities.}

For triangle with points A, B, C and corresponding angles a, b, c:

\begin{equation}
\frac{sin(a)}{\overline{BC}} = \frac{sin(b)}{\overline{CA}} = \frac{sin(c)}{\overline{AB}}
\end{equation}

\section{Additional Notes}


{\color{blue} FREQUENCY IS THE FUCKING NUMBER OF CYCLES PER UNIT LENGTH, SO THE FUCKING NUMBER OF CYCLES IN A DISTANCE IS THE FUCKING FREQUENCY TIMES THE FUCKING DISTANCE. THEN, TO FUCKING CONVERT CYCLES TO RADIANS (WHICH YOU HAVE TO DO TO BE ABLE TO TALK ABOUT THE PHASE OF THE FUCKING WAVE) YOU JUST FUCKING MULTIPLY BY $2\pi$.\\

Frequency is cycles per unit distance.

Number of cycles in a distance is frequency times distance.

Amount of spatial phase in that distance is cycles * $2\pi.$

Because $k$ is both $2\pi$ and the frequency in one variable, just multiply by $k$.
}

{\color{red} 1.22 lambda d equation}

{\color{red} Energy of photon}

Going to higher indices of refraction means rays bend towards the normal.

Def of dialectric


Right before the exam:

Calculating numerical aperture (chapter 1)
        
\end{document}
